Question

I am having trouble building a regular expression with the set of strings over {a, b, c} that is an odd length with exactly one a. Here is my best attempt so far:

(bb|bc|cb|cc)*a(bb|bc|cb|cc)*

This does good for even b and c on either side of the a, but does not account for a odd b and c combination on either side of the a.

Any hints?

Answer 1

Your string will be a prefix followed by a followed by a suffix.

Both prefix and suffix can be zero length. If not, they have to be either both even or both uneven. This means you have two main cases.

EVENPREFIX a EVENSUFFIX | UNEVENPREFIX a UNEVENSUFFIX

Try this (incomplete and wrong):

([bc][bc])*a([bc][bc])*|([bc][bc][bc])*a([bc][bc][bc])*

There is still one uneven case missing: a single [bc]:

(([bc][bc])*a([bc][bc])*)|([bc]([bc][bc])*a[bc]([bc][bc])*)

According to http://www.fileformat.info/tool/regex.htm, this matches

• a
• cac
• ccabb

I expect it matches the rest too…

The left side guarantees even (or empty) sequences of b or c.
The right side is either a single b or c followed by a multiple of two (so that it stays uneven).

Kobi came up with this refinement of the above:

([bc][bc])*(a|[bc]a[bc])([bc][bc])*

How does this work?

The first group is guaranteed to be even. The second group is guaranteed to be uneven with a single a inside. The third group is guaranteed to be be even. Thus, the whole is guaranteed to be uneven.